# Dynamic Programming From An

## Presentation on theme: "Dynamic Programming From An"— Presentation transcript:

Dynamic Programming From An
Excel Perspective

Dynamic Programming From An Excel Perspective
Ranette Halverson, Richard Simpson Catherine Stringfellow Department of Computer Science Midwestern State University

Dynamic Programming From An Excel Perspective
A popular method for solving problems by breaking them down into overlapping sub-problems that display optimal substructure Can be thought of as a top-down approach utilizing a bottom-up evaluation Normally used to solve optimization problems

Dynamic Programming From An Excel Perspective
Generally taught in the freshman application classes Seldom taught to computer science majors In reality CS majors need to be able to use spreadsheets So what do we do?

Dynamic Programming From An Excel Perspective
Solution Do not need to teach the spreadsheet AT ALL Include spreadsheet usage in a few of their projects and/or homework Spreadsheet usage includes Graphing data collected via empirical analysis of two algorithms. Rapidly construct mathematical tables for applications Simulating wave-front parallel algorithms Evaluating dynamic programming tables (the point of this talk)

Dynamic Programming From An Excel Perspective
A Very Simple Example (used in Computer Science for Science Majors) The memo-ization of the recursive Fibonacci function. Remember the complexity of the following? int Fib( int n) { if (n<3) return 1 else return ( Fib(n-1)+Fib(n-2) ); }

Dynamic Programming From An Excel Perspective
Two well-known O(n) solutions int FibMemo(int n,int * A){ if (A[n]!=0) return A[n]; else { A[n]= FibMemo(n-1,A) + FibMemo(n-2,A); return A[n]; } }; int Fib(int n) { int * A = new int[n+1] ; for (int i=1;i<n+1;i++){ A[i]=0;} A[1]=A[2]=1; return FibMemo(n,A); } // A recursive Memoized version int Fib( int n) { A=new int[n+1]; A[1]=A[2]=1; for(int i=3 ; i<=n ; i++) A[i] = A[i-1] + A[i-2]; return A[n]; } // Pure Bottom up calculation using // an array. The non array version is // not relative to our discussion.

Dynamic Programming From An Excel Perspective
Excel’s simple approach Kand copy cell to the right =A1+B1

Dynamic Programming From An Excel Perspective
Pascal's triangle is constructed in Excel in the bottom-up approach. The programmed solution can be handled via DP as in the Fibonacci example, either using an array with or without memoized recursion. The pure recursive version is computationally unacceptable. formula =B1+A2 is copied from B2 to the remaining cells.

Dynamic Programming From An Excel Perspective
There are many DP algorithms that appear throughout our curriculum. Longest Common Subsequence Bioinformatics class. Sequence Alignment: Bioinformatics Optimal Binary Search Tree: Algorithm Analysis Matrix Chain Multiplication: Algorithms Analysis Graph Algorithms

Dynamic Programming From An Excel Perspective
Longest Common Subsequence (LCS) Definition: find the longest subsequence that is common to two (or more) sequences. Example Seq1 = B D C A B A Seq2 = A B C B D A B LCS = BCBA Note: The LCS is not a substring!

Dynamic Programming From An Excel Perspective
Longest Common Subsequence (LCS) DP leads to the following recursive approach. Let z=z1 z2 … xk be the LCS of x1 x2 … xi-1 xi y1 y2 … yj-1 yj 𝑐 𝑖,𝑗 = 0 𝑖𝑓 𝑥=0 𝑜𝑟 𝑗=0 𝑐 𝑖−1,𝑗−1 +1 𝑖𝑓 𝑖,𝑗>0 𝑎𝑛𝑑 𝑥 𝑖 = 𝑦 𝑗 max(𝑐 𝑖,𝑗−1 ,𝑐 𝑖−1,𝑗 𝑖𝑓 𝑖,𝑗>0 𝑎𝑛𝑑 𝑥 𝑖 ≠ 𝑦 𝑗 Where c[ib ,j] is the length of the LCS of x1..xi and y1..yj

Dynamic Programming From An Excel Perspective
The initialized LCS Table Copy the following cell formula to all grey cells. These represent the c(i,j)’s IF(D\$2=\$B4,C3+1,MAX(D3,C4)) Cell formula

Dynamic Programming From An Excel Perspective
And the solution is Note diagonal increments Length of LCS

Dynamic Programming From An Excel Perspective
DP Problems with complex table manipulation Optimal Binary Search Tree (in paper) Matrix Chain Multiplication Question: What do you do with problems that require the processing of rows or columns in such a way that the usual cell function approach is not adequate? Excel does not allow cell function side effects! Hmm. Write the code in the include macro language (VB?)

Dynamic Programming From An Excel Perspective
Summarizing CS students can benefit from work with Excel Excel can support many projects in the CS curriculum. Table processing available in Excel supports some algorithm visualization quite well. This approach works particularly well with the simpler DP problems.

THE END